CHAPTER 2. ANALYTICAL TOOLS joint production and arbitrary returns to scale. If we rule out joint produc tion, it is often more convenient to work with the cost function. It is defined as follows C(w, Mi)=minfwvlf(a)=xi1. where v is the vector of factors used to produced good j and f is the pro- duction function. In addition, you should know and prove that with c.r.s C(w, i)=b(w)ci, where b() is the unit cost function for good j. We will work with b instead of c, so it is useful to list its properties(prove them b(w) is concave in w The optimum choice of input coefficients a'is given by a(w)=bi(a) b(w) is homogenous of degree one in w and thus aw=b The optimal choice of inputs to produce is given by y(w)=a(a), There is a connection between the cost and the revenue functions. which should not surprise you as it comes from duality r(p, v)=minal Vj: b()2Pil Thus the revenue function can alternatively defined as the value function for a problem where we minimize factor payments when unit costs are at least as large as output prices. For proof, see DN Ch 2, p 45. It is enough to note that for both representations of the revenue function we can write down the Kuhn-Tucker sufficient conditions. We will alwavs assume that all factors are fully emploved. so that aij an vj:b(x)≥n,x;≥0,[b()-pjx;=0 The second condition allows for the possibility that not all goods are actually produced, and we will see that happen in many important cases
CHAPTER 2. ANALYTICAL TOOLS 8 joint production and arbitrary returns to scale. If we rule out joint production, it is often more convenient to work with the cost function. It is defined as follows: c j (w, xj ) = min v j {wvj | f j (v j ) = xj}, where v j is the vector of factors used to produced good j and f is the production function. In addition, you should know and prove that with c.r.s. c j (w, xj ) = b j (w)xj , where b j (w) is the unit cost function for good j. We will work with b j instead of c j , so it is useful to list its properties (prove them!): • b j (w) is concave in w. • The optimum choice of input coefficients a j is given by a j (w) = b j w(w). • b j (w) is homogenous of degree one in w and thus a jw = b j . • The optimal choice of inputs to produce xj is given by v j (w) = a j (w)xj . There is a connection between the cost and the revenue functions, which should not surprise you as it comes from duality: r(p, v) = min w {wv| ∀j : b j (w) ≥ pj}. Thus the revenue function can alternatively defined as the value function for a problem where we minimize factor payments when unit costs are at least as large as output prices. For proof, see DN Ch.2, p.45. It is enough to note that for both representations of the revenue function we can write down the Kuhn-Tucker sufficient conditions. We will always assume that all factors are fully employed, so that: X j a j ixj = vi , and ∀j : b j (w) ≥ pj , xj ≥ 0, [b j (w) − pj ]xj = 0. The second condition allows for the possibility that not all goods are actually produced, and we will see that happen in many important cases
CHAPTER 2. ANALYTICAL TOOLS 2.3 Consumer choice We will represent consumers' choice mainly by the expenditure function e(p, u)=minpcl f(c)>u where f(c) is now the utility function and c is the consumption vector. The problem is mathematically the same as cost minimization, so we can just list the properties of the expenditure function as follows e(p, u)is increasing and concave in p If e is differentiable in p, then c(p, u)=ep(c, u) e(p, u) is linearly homogenous in p, and thus pep=e(p, u) If preferences are homothetic, e(p, u)=e(p)u. e(p) is also called the true price inder, because it gives the required expenditure to buy one unit of utility. We will use it a lot later epp(p, a)is negative semidefinite, and epp(p, u)p=cp(p, u)p=0 .(P1-P2(C1-c2)<0-compensated demand functions are downward sloper The expenditure function gives us the compensated demand function but in many cases(for example when doing comparative statics)we need the uncompensated one. There is well-known connection between the two, and it leads to the following properties c (p, u)=dpp, e(p, u)]: compensated and regular demand equal each other if the income level -y -is given by the expenditure function evaluated at a Cp(p, u)=dp(p,y)+dy(p, y)d(p, y)(the Slutsky-Hicks equation) d,(p, y)=epu(p, u)/eu(p, u), where y=e(p, u)
CHAPTER 2. ANALYTICAL TOOLS 9 2.3 Consumer choice We will represent consumers’ choice mainly by the expenditure function: e(p, u) = min c {p c| f(c) ≥ u}, where f(c) is now the utility function and c is the consumption vector. The problem is mathematically the same as cost minimization, so we can just list the properties of the expenditure function as follows: • e(p, u) is increasing and concave in p. • If e is differentiable in p, then c(p, u) = ep(c, u). • e(p, u) is linearly homogenous in p, and thus p ep = e(p, u). • If preferences are homothetic, e(p, u) = ¯e(p) u. ¯e(p) is also called the true price index, because it gives the required expenditure to buy one unit of utility. We will use it a lot later. • epp(p, u) is negative semidefinite, and epp(p, u)p = cp(p, u)p = 0. • (p1 − p2)(c1 − c2) ≤ 0 – compensated demand functions are downward sloping. The expenditure function gives us the compensated demand function, but in many cases (for example when doing comparative statics) we need the uncompensated one. There is well-known connection between the two, and it leads to the following properties: • c(p, u) = d[p, e(p, u)]: compensated and regular demand equal each other if the income level – y – is given by the expenditure function evaluated at u • cp(p, u) = dp(p, y) + dy(p, y)d(p, y) T (the Slutsky-Hicks equation). • dy(p, y) = epu(p, u)/eu(p, u), where y = e(p, u)
CHAPTER 2. ANALYTICAL TOOLS 2. 4 The Meade utility functions A useful tool is the Meade(or direct trade) utility function that condenses the information found in the various envelope functions. It is particularly useful for analyzing the effects of tariffs and for normative purposes. We will not use it much, but some of the literature does, so you should be familiar with it. It is defined as follows: p(m, u)=maxf(a +m)l (a,U)EYJ, where the notation is as before. In particular, y is a convex production set and f is a quasi-concave utility function. Thus (m, v) shows the maximum utility when production is feasible, factor endowments are given by v and the import vector is m. In essence we"optimize out"the production vector to concentrate on net trade and endowments We used this construct -without mentioning its name -in the pure exchange model at the beginning As before, we can list the properties of o as follows o(m, a)is increasing in(m, v)(obvious) o(m, v) is quasi-concave in m. Let aI and a2 be the optimal plans corresponding to mI and m2. Since Y is convex, 1/2(1+r2)is fea- sible. Then (1/2(m1 +m2),u>f[1/(1+12)+1/ 2(m1 +m2) f1/2(x1+m1)+1/2( ≥min{f( mino(m1, u),o(m2,U)1 m(, v)o p- the gradient of o w.r.t. m is proportional to prices (Envelope Theorem and FOC of competitive equilibrium) o(m,v)∝u Actually there is another trade utility function, which is called the indirect rade utility function. It is defined as follow H(p,b,)=max{f(c)|pc≤r(p,v)-b,c≥ It gives the maximum utility that an economy can attain given prices, factor endowments and trade balance(which does not have to be restricted to 0) Since dn does not use it, we will not either, but you should know that it exists I You can learn more from the following paper. A D. Woodland: Direct and indirec trade utility functions, The Review of Economic Studies, Oct. 1980
CHAPTER 2. ANALYTICAL TOOLS 10 2.4 The Meade utility functions A useful tool is the Meade (or direct trade) utility function that condenses the information found in the various envelope functions. It is particularly useful for analyzing the effects of tariffs and for normative purposes. We will not use it much, but some of the literature does, so you should be familiar with it. It is defined as follows: φ(m, v) = max x {f(x + m)|(x, v) ∈ Y }, where the notation is as before. In particular, Y is a convex production set and f is a quasi-concave utility function. Thus φ(m, v) shows the maximum utility when production is feasible, factor endowments are given by v and the import vector is m. In essence we “optimize out” the production vector to concentrate on net trade and endowments. We used this construct – without mentioning its name – in the pure exchange model at the beginning. As before, we can list the properties of φ as follows: • φ(m, v) is increasing in (m, v) (obvious). • φ(m, v) is quasi-concave in m. Let x1 and x2 be the optimal plans corresponding to m1 and m2. Since Y is convex, 1/2(x1 + x2) is feasible. Then φ[1/2(m1 + m2), v] ≥ f[1/2(x1 + x2) + 1/2(m1 + m2)] = f[1/2(x1 + m1) + 1/2(x2 + m2)] ≥ min{f(x1 + m1), f(x2 + m2)} = min{φ(m1, v), φ(m2, v)}. • φm(m, v) ∝ p – the gradient of φ w.r.t. m is proportional to prices (Envelope Theorem and FOC of competitive equilibrium). • φv(m, v) ∝ w. Actually there is another trade utility function, which is called the indirect trade utility function. It is defined as follows: H(p, b, v) = max c {f(c)| p c ≤ r(p, v) − b, c ≥ 0}. It gives the maximum utility that an economy can attain given prices, factor endowments and trade balance (which does not have to be restricted to 0). Since DN does not use it, we will not either, but you should know that it exists.1 1You can learn more from the following paper. A.D. Woodland: Direct and indirect trade utility functions, The Review of Economic Studies, Oct. 1980
Chapter 3 Equilibrium and the gains from rade 3.1 Defining the equilibrium Here we will establish some basic properties of the international equilibrium In most cases we will assume a representative consumer and fixed factor supply. The latter can be relaxed fairly easily, but in most of the literature it is not dn deals with the flexible factor supply case, so you can take a look there. About the former, heterogeneity is interesting when we look at gains from trade(see below) and it can be managed fairly easily. In most other cases, however, we have to revert to the representative consumer assumption The problem is, of course, that we can say very little about aggregate demand functions for a general utility function, because of the aggregation problem Thus we need to make the heroic assumption of a representative consumer Sometime we even have to go further, and assume homothetic preferences will remind you when this is the case Let us write down the conditions for autarchy. Using the revenue and xpenditure functions, it is an easy task e(p, u)=r(p, u) ep(p, u)=p(p, (3 The first equation is the identity of GDP and national income. The second is actually a vector equation, and it gives us market-clearing conditions for all goods. We know by Walras law that one equation is redundant and that
Chapter 3 Equilibrium and the gains from trade 3.1 Defining the equilibrium Here we will establish some basic properties of the international equilibrium. In most cases we will assume a representative consumer and fixed factor supply. The latter can be relaxed fairly easily, but in most of the literature it is not. DN deals with the flexible factor supply case, so you can take a look there. About the former, heterogeneity is interesting when we look at gains from trade (see below) and it can be managed fairly easily. In most other cases, however, we have to revert to the representative consumer assumption. The problem is, of course, that we can say very little about aggregate demand functions for a general utility function, because of the aggregation problem. Thus we need to make the heroic assumption of a representative consumer. Sometime we even have to go further, and assume homothetic preferences. I will remind you when this is the case. Let us write down the conditions for autarchy. Using the revenue and expenditure functions, it is an easy task: e(p, u) = r(p, v) ep(p, u) = rp(p, v). (3.1) The first equation is the identity of GDP and national income. The second is actually a vector equation, and it gives us market-clearing conditions for all goods. We know by Walras’ law that one equation is redundant and that 11
CHAPTER 3. EQUILIBRIUM AND THE GAINS FROM TRADE we can normalize the price of one good. We will specify which one when necessary The free trade equilibrium is similarly easy to characterize. Let us keep our convention of using upper-case letters for foreign variables, then we have e(p, u)=r(p, a E(p, U)=R(p, V) (32) ep(p, u)+ Ep(p, U)=Tp(p, v)+Rp(p, V) We can easily relax the assumption of a representative consumer. Let h ndex consumers and assume that each of them owns yh amount of factors Then, recalling that factor prices are given by ru, we have (p, U)=Ry(p, v)V ∑"(p,n)+∑E(p,U)=p)+B(V) Notice that with identical homothetic utility functions there is a well-defined aggregate demand function that takes the same form as the individual de mand functions 3.2 Gains from trade Let us start with the representative consumer case. Here a simple revealed preference argument shows that there are gains from trade. We dont in fact have to use the equilibrium conditions, just compare utility at autarchy and free trade prices(pa and p). The argument is as follows e(p2,l2)≤p2c P (p2,u2) Since e(p, u) is an increasing function of u, utility at free trade must be at least as high as in autarchy. Notice that there are actually two inequalities
CHAPTER 3. EQUILIBRIUM AND THE GAINS FROM TRADE 12 we can normalize the price of one good. We will specify which one when necessary. The free trade equilibrium is similarly easy to characterize. Let us keep our convention of using upper-case letters for foreign variables, then we have the following: e(p, u) = r(p, v) E(p, U) = R(p, V ) (3.2) ep(p, u) + Ep(p, U) = rp(p, v) + Rp(p, V ). We can easily relax the assumption of a representative consumer. Let h index consumers and assume that each of them owns v h amount of factors. Then, recalling that factor prices are given by rv, we have: e h (p, uh ) = rv(p, v)v h E H(p, U H) = RV (p, V )V H (3.3) X h e h p (p, uh ) +X H E H p (p, U H) = rp(p, v) + Rp(p, V ). Notice that with identical homothetic utility functions there is a well-defined aggregate demand function that takes the same form as the individual demand functions. 3.2 Gains from trade Let us start with the representative consumer case. Here a simple revealed preference argument shows that there are gains from trade. We don’t in fact have to use the equilibrium conditions, just compare utility at autarchy and free trade prices (p a and p t ). The argument is as follows: e(p t , ua ) ≤ p t c a = p tx a ≤ r(p t , v) = e(p t , ut ) Since e(p, u) is an increasing function of u, utility at free trade must be at least as high as in autarchy. Notice that there are actually two inequalities